Background/Objectives: Antibiotic resistance in bacteria poses a major health problem worldwide. Therefore, to counteract this life-threatening problem, we sought to investigate in the present study the possible potentiation of the efficacy of amoxicillin (AMX) and clavulanic acid (CA) by 1,8-cineole (CN), a candidate resistance-modulating agent. The approach seeks to investigate, in vitro and in silico, the interactions among these three molecules. Methods: The antibacterial activity was determined against resistant Escherichia coli (E. coli) using microdilution methods, synergy tests, and time-kill assays for AMX, CA, and CN, used either separately or in combinations: AMX–CA, AMX–CN, CA–CN, and AMX–CA–CN. Furthermore, an in silico drug design methodology was employed, utilizing an integrated workflow that combines Density Functional Theory (DFT) for ligand optimization with molecular docking simulations to evaluate binding energies and interactions between the penicillin-binding protein (PBP) and ligands. Results: In vitro synergy experiments and time-kill assays revealed substantial antibacterial efficacy of the AMX–CA–CN combination. In silico analyses, performed under the simplifying assumption of a pre-assembled multi-ligand entity, were consistent with these findings: within our docking model, the AMX–CA–CN combination exhibited the most favorable computed binding affinity to a representative penicillin-binding protein (PBP3, PDB 7ONW). Conclusions: 1,8-cineole can potentiate the antibacterial effects of AMX–CA, indicating that the AMX–CA–CN combination warrants further evaluation as a candidate adjunctive strategy against ESBL-producing E. coli.